Strichartz and local smoothing estimates for the fractional Schr\"odinger equations over fractal time

TL;DR

This paper derives Strichartz and local smoothing estimates for fractional Schrödinger equations over fractal time sets, extending classical results to fractal contexts using Assouad dimension concepts.

Contribution

It introduces new Strichartz estimates for fractional Schrödinger operators over fractal time sets, incorporating fractal dimensions into the analysis.

Findings

Established Strichartz estimates over fractal sets with Assouad dimension.

Proved $L^2$ local smoothing estimates for fractional Schrödinger equations.

Extended classical estimates to rough fractal time domains.

Abstract

We obtain Strichartz-type estimates for the fractional Schr\"odinger operator $f \mapsto e^{it(-\Delta)^{\gamma/2}} f$ over a time set $E$ of fractal dimension. To obtain those estimates capturing fractal nature of $E$, we employ the notions in the spirit of the Assouad dimension, such as, bounded Assouad characteristic and Assouad specturm. We also prove the estimate $$ \| e^{it(-\Delta)^{\gamma/2}} f \|_{L_t^q(\mathrm{d}\mu; L_x^r(\mathbb{R}^d))} \le C \|f\|_{H^s}, $$ where $\mu$ is a measure satisfying an $\alpha$-dimensional growth condition. In addition, we establish related inhomogeneous estimates and $L^2$ local smoothing estimates. A surprising feature of our work is that, despite dealing with rough fractal sets, we extend the known estimates for the fractional Schr\"odinger operators in a natural way, precisely consistent with the associated fractal dimensions.